I'm on lesson 2, is the student supposed to continue to work on the same essay he did in lesson 1?

Yes, and no. Each lesson will have its own essay. Since you are on Lesson 2, your students will start this cycle with a new issue, a new ANI, and then the new invention idea for this lesson-the 5 Common Topics. They will use this new, Lesson 2 invention, work for the 2 essay of the program.

No, because you will be adding to the practice essay. See the explanation on the practice essay.